Abstract:

We will study different properties of eigenvectors of random matrices using the dynamical method. In the first part of the talk, I will discuss the distribution and the quantum unique ergodicity property of eigenvectors of deformed Wigner matrices. We will see that eigenvector entries are asymptotically Gaussian with a specific variance localizing them on a small part of the spectrum. The proof relies on a priori local laws for this model and the eigenvector moment flow. In the second part of the talk, I will give some insight in the study of the fluctuations of eigenvectors through the introduction of a new observable following the same eigenvector moment flow.